When allowing for \(K_{i} \ne 0\) , it is intuitively apparent that
having reached point \(x(t_{s}) \in S_{0}\)  at \(t = t_{s}\) , the
trajectory cannot remain there for all times \(t_{s} < t < \infty \) .
While for the motion states \(x_{3}(t_{s})=0\)  and \(x_{2}(t_{s}) =\mathrm {const} \ne 0\) , the integral control effort \(K_{i}x_2(t_{s}) \int ^{t}_{t_{s}} dt\)  grows continuously and, at some
finite time (\(t_{c} > t_{s}\) ), should lead to the breakaway
[1]} and a new onset of continuous motion. This
alternating phase, upon system sticking, is often referred to as
slipping, cf., e.g., [2]}, so that a
stick-slip motion [3]}, [4]} appears, also in the
form of limit cycles. In order to analyze the friction-induced
limit cycles, sometimes denoted as hunting-limit cycles, cf.
[5]}, we firstly need to look into the system
dynamics during stiction, i.e., for \(t_{s} < t < t_{c}\) . Recall
that in this phase, the system (REF )-(REF ) yields as
continuously switching, owing to \(x_{3} = 0\)  and to the
discontinuous relay nonlinearity in the feedback loop. The
developments given below are motivated by analysis of the
existence of fast switches, provided in [6]} for
RFS, while the obtained results rely on the sliding-mode
principles, see, e.g., [7]}, [8]}.
